We present a general central limit theorem with simple, easy-to-check covariance-based sufficient conditions for triangular arrays of random vectors when all variables could be interdependent. The result is constructed from Stein's method, but the conditions are distinct from related work. We show that these covariance conditions nest standard assumptions studied in the literature such as $M$-dependence, mixing random fields, non-mixing autoregressive processes, and dependency graphs, which themselves need not imply each other. This permits researchers to work with high-level but intuitive conditions based on overall correlation instead of more complicated and restrictive conditions such as strong mixing in random fields that may not have any obvious micro-foundation. As examples of the implications, we show how the theorem implies asymptotic normality in estimating: treatment effects with spillovers in more settings than previously admitted, covariance matrices, processes with global dependencies such as epidemic spread and information diffusion, and spatial process with Mat\'{e}rn dependencies.

翻译：我们提出了一个一般的中心极限定理，该定理针对所有变量可能相互依赖的三角阵列随机向量，给出了基于协方差的简单、易于检验的充分条件。该结果基于Stein方法构建，但条件与现有研究有所不同。我们证明，这些协方差条件涵盖了文献中研究的标准假设，例如$M$-相依、混合随机场、非混合自回归过程以及依赖图，而这些假设本身并不相互蕴含。这使得研究者能够基于整体相关性使用高层次但直观的条件，而非更复杂且具有限制性的条件（例如随机场中可能缺乏明显微观基础的强混合条件）。作为应用的例证，我们展示了该定理如何推导出以下估计中的渐近正态性：在比以往更广泛的场景中具有溢出效应的处理效应估计、协方差矩阵估计、具有全局依赖性的过程（如流行病传播与信息扩散）估计，以及基于Matérn依赖性的空间过程估计。